Math Problem Statement

Write the system of equations as an augmented matrix.

Solution

The problem asks us to write the given system of equations in augmented matrix form. Let's analyze the equations:

  1. c−3m=300c - 3m = 300
  2. 3c−11m+b=1503c - 11m + b = 150
  3. −3m+b=400-3m + b = 400

Step-by-Step Solution:

To create the augmented matrix:

  1. Arrange the equations in standard form: coefficients of cc, mm, and bb (if present) on the left-hand side, with constants on the right-hand side.
  2. Use the coefficients of the variables as matrix entries, including zeros for missing variables.
  3. Place the constants on the right-hand side as the last column.

Standardized Equations:

  • Equation 1: c−3m+0b=300c - 3m + 0b = 300 → Coefficients: [1,−3,0∣300][1, -3, 0 | 300]
  • Equation 2: 3c−11m+b=1503c - 11m + b = 150 → Coefficients: [3,−11,1∣150][3, -11, 1 | 150]
  • Equation 3: 0c−3m+b=4000c - 3m + b = 400 → Coefficients: [0,−3,1∣400][0, -3, 1 | 400]

Augmented Matrix:

1 & -3 & 0 & 300 \\ 3 & -11 & 1 & 150 \\ 0 & -3 & 1 & 400 \end{bmatrix}$$ ### Final Answer: The augmented matrix for this system is: $$\boxed{ \begin{bmatrix} 1 & -3 & 0 & 300 \\ 3 & -11 & 1 & 150 \\ 0 & -3 & 1 & 400 \end{bmatrix} }$$ Would you like me to explain further or assist with solving the system of equations? --- ### Related Questions: 1. What is the purpose of augmented matrices in solving linear systems? 2. How can this matrix be solved using row-reduction (Gaussian elimination)? 3. What are other methods for solving this system besides matrices (e.g., substitution)? 4. How would you interpret the solution geometrically if $$c$$, $$m$$, and $$b$$ represent variables in 3D space? 5. How can software like MATLAB or Python help solve such matrix problems? --- ### Tip: When creating an augmented matrix, always ensure the variables are ordered consistently across all equations!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Augmented Matrices
Systems of Linear Equations

Formulas

Matrix Representation of Linear Systems

Theorems

Matrix Row Operations
Gaussian Elimination

Suitable Grade Level

Grades 9-12

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